Mathematical Analysis of Four Fundamental Epidemiological Models for Monkeypox Disease Outbreaks: On the Pivotal Role of Human–Animal Order Parameters—In Memory of Hermann Haken

Four fundamental models that describe the spread of Monkeypox disease are analyzed: the SIR-SIR, SEIR-SIR, SIR-SEIR, and SEIR-SEIR models. They form the basis of most Monkeypox diseases models that are currently discussed in the literature. It is shown that the way the model subpopulations are organized in disease outbreaks and evolve relative to each other is determined by the relevant unstable system eigenvectors, also called order parameters. For all models, analytical expressions of the order parameters are derived. Under appropriate conditions these order parameters describe the initial outbreak phases of exponential increase in good approximation. It is shown that all four models exhibit maximally two order parameters and maximally one human–animal order parameter. The human–animal order parameter firmly connects the outbreak dynamics in the animal system with the dynamics in the human system. For the special case of the SIR-SIR model, it is found that the two possible order parameters completely describe the dynamics of infected humans and animals during entire infection waves. Finally, a simulation of a Monkeypox infection wave illustrates that in line with the aforementioned analytical results the leading order parameter explains most of the variance in the infection dynamics.